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Story behind Story Problems 1

The Real Story behind Story Problems:

Effects of Representations on Quantitative Reasoning

Kenneth R. Koedinger*

Carnegie Mellon University

Mitchell J. Nathan

University of Colorado

RUNNING HEAD: Story behind Story Problems

*Contact Information:

Ken KoedingerHuman-Computer Interaction InstituteCarnegie Mellon UniversityPittsburgh, PA 15213Phone: 412-268-7667Fax: 412-268-1266Email: [email protected]

mailto:[email protected]


ABSTRACT

We explored how differences in problem representations change both the performance and underlying

cognitive processes of beginning algebra students engaged in quantitative reasoning. Contrary to beliefs

held by practitioners and researchers in mathematics education, we found that students were more

successful solving simple algebra story problems than solving mathematically equivalent equations.

Contrary to some views of situated cognition, this result is not simply a consequence of situated world

knowledge facilitating problem solving performance, but rather a consequence of student difficulties

with comprehending the formal symbolic representation of quantitative relations. We draw on analyses

of students’ strategies and errors as the basis for a cognitive process explanation of when, why, and how

differences in problem representation affect problem solving. In general, we conclude that differences in

external representations can affect performance and learning when one representation is easier to

comprehend than another or when one representation elicits more reliable and meaningful solution

strategies than another.

Key words: problem solving, knowledge representation, mathematics learning, cognitive processes,

complex skill acquisition


INTRODUCTION

Story Problems Are Believed to Be Difficult

A commonly held belief about story problems at both the arithmetic and algebra levels is that they are

notoriously difficult for students. Support for this belief can be seen among a variety of populations

including the general public, textbook authors, teachers, mathematics education researchers, and

learning science researchers. For evidence that this belief is commonly held within the general public,

ask your neighbor. More likely than not, he or she will express a sentiment toward story problems

along the lines of Gary Larson’s cartoon captioned "Hell's Library" that contains book shelves full of

titles like "Story Problems," "More Story Problems," and "Story Problems Galore." That many

textbook authors believe in the greater difficulty of story problems is supported by an analysis of

textbooks by Nathan, Long, and Alibali (2002). In 9 of the 10 textbooks they analyzed, new topics are

initially presented through symbolic activities and only later are story problems presented, often as

“challenge problems”. The choice of this ordering is consistent with the belief that symbolic

representations are more accessible to students than story problems.

More direct evidence of the common belief in the difficulty of story problems comes from surveys

of teachers and mathematics educators. In a survey of 67 high school mathematics teachers, Nathan &

Koedinger (2000a) found that most predicted that story problems would be harder for algebra students

than matched equations. Nathan & Koedinger (2000a) also surveyed 35 mathematics education

researchers. The majority of these researchers also predicted that story problems would be harder for

algebra students than matched equations. In another study of 105 K-12 mathematics teachers, Nathan &

Koedinger (2000b) found that significantly more teachers agree than disagree with statements like

“Solving math problems presented in words should be taught only after students master solving the

same problems presented as equations.” This pattern was particularly strong among the high school

teachers in the sample (n = 30).


Belief in the difficulty of story problems is also reflected in the learning science literature. Research

on story problem solving, at both the arithmetic (Carpenter, Kepner, Corbitt, Lindquist, & Reys, 1980;

Cummins et al., 1988; Kintsch & Greeno, 1985) and algebra levels (Clement, 1982; Nathan et al., 1992;

Paige & Simon, 1966), has emphasized the difficulty of such problems. For instance, Cummins and her

colleagues (1988, p. 405) commented "word problems are notoriously difficult to solve". They

investigated first graders’ performance on matched problems in story and numeric format for 18

different categories of one operator arithmetic problems. Students were 27% correct on the “Compare

2” problem in story format: “Mary has 6 marbles. John has 2 marbles. How many marbles does John

have less than Mary?” but were 100% correct on the matched numeric format problem, “6 - 2 = ?”.

They found performance on story problems was worse than performance on matched problems in

numeric format for 14 of the 18 categories and was equivalent for the remaining 4 categories. Belief in

the greater difficulty of story problems is also evident in the broader developmental literature. For

instance, Geary (1994, p. 96) states "children make more errors when solving word problems than when

solving comparable number problems."

Although the research that Cummins and others (1988) performed and reviewed addressed

elementary-level arithmetic problem solving, they went on to make the broader claim that "as students

advance to more sophisticated domains, they continue to find word problems in those domains more

difficult to solve than problems presented in symbolic format (e.g., algebraic equations)" (p. 405).

However, apart from our own studies reported here, this broader claim appears to have remained

untested (cf., Reed, 1998). We have not found prior experimental comparisons of solution correctness

on matched algebra story problems and equations for students learning algebra. In a related study,

Mayer (1982a) used solution times to make inferences about the different strategies that well-prepared

college students use on algebra word and equation problems. He found a different profile of solution

times for word problems than equations as problems varied in complexity and accounted for these

differences by the hypothesis that students use a goal-based “isolate” strategy on equations and a less


memory-intensive “reduce” strategy on word problems. Overall, students took significantly longer to

solve 1-5 step word problems (about 15 seconds) than matched equations (about 5 seconds) with no

reliable difference in number of errors (7% for word problems, 4% for equations). Whereas Mayer’s

study focused on timing differences for well-practiced participants, the studies reported here focus on

error differences for beginning algebra students.

Why are Story Problems Difficult?

What might account for the purported and observed difficulties of story problems? As many researchers

have observed (Cummins et al., 1988; Hall et al., 1989; Lewis & Mayer, 1987; Mayer, 1982b), the

process of story problem solving can be divided into a comprehension phase and a solution phase (see

Figure 1). In the comprehension phase, problem solvers process the text of the story problem and create

corresponding internal representations of the quantitative and situation-based relationships expressed in

that text (Nathan, Kintsch, & Young, 1992). In the solution phase problem solvers use or transform the

quantitative relationships that are represented both internally and externally to arrive at a solution. Two

kinds of process explanations for the difficulty of story problems correspond with these two problem-

solving phases. We will return to these explanations, but first we describe how these two phases interact

during problem solving (for more detail see Koedinger & MacLaren, 2002).

----- Insert Figure 1 about here -----

In general, the comprehension and solution phases are typically interleaved rather than

performed sequentially. Problem solvers iteratively comprehend first a small piece of the problem

statement (e.g., a clause or sentence) and then produce a piece of corresponding external representation

(e.g., an arithmetic operation or algebraic expression), often as an external memory aid. In Figure 1, the

double-headed arrows within the larger arrows are intended to communicate this interactivity. During

problem solving, aspects of newly constructed internal or external representations may influence further

comprehension in later cycles (Kinstch, 1998). For example, after determining that the unknown value


is the number of donuts, the reader may then search for and reread a clause that uses number of donuts

in a quantitative relation. Similarly, the production of aspects of the external representation may help

maintain internal problem-solving goals that, in turn, may direct further comprehension processes.

A number of researchers have provided convincing evidence that errors in the comprehension

phase well account for story problem solving difficulties (e.g., Cummins et al., 1988; Lewis & Mayer,

1987). For instance, Cummins et al. (1988) demonstrated that variations in first graders’ story problem

performance were well predicted by variations in problem recall and that both could be accounted for by

difficulties students had in comprehending specific linguistic forms like “some”, “more X’s than Y’s”,

and “altogether”. They concluded that “text comprehension factors figure heavily in word problem

difficulty” (p. 435). Lewis and Mayer (1987) summarized past studies with K-6 graders and their own

studies with college students showing more solution errors on arithmetic story problems with

“inconsistent language” (e.g., when the problem says “more than”, but subtraction is required to solve

it) than problems with consistent language. Teachers’ intuitions about the difficulty of algebra story

problems (c.f., Nathan & Koedinger, 2000) appear to be in line with these investigations of

comprehension difficulties with arithmetic story problems. As one teacher explained “students are used

to expressions written algebraically and have typically had the most practice with these … translating

‘English’ or ‘non-mathematical’ words is a difficult task for many students”.

A second process explanation for the difficulty of story problems focuses on the solution phase

and particularly on the strategies students use to process aspects of the problem. A common view of

how story problems are or should be solved, particularly at the algebra level, is that the problem text is

first translated into written symbolic form and then the symbolic problem is solved (e.g., see Figure 4a).

If problem solvers use this translate-and-solve strategy, then clearly story problems will be harder than

matched symbolic problems since solving the written symbolic problem is an intermediate step in this

case.


At least at the algebra level, the translate-and-solve strategy has a long tradition as the

recommended approach. Paige & Simon (1966) comment regarding an algebra-level story problem,

“At a common-sense level, it seems plausible that a person solves such problems by, first, translating the

problem sentences into algebraic equations and, second, solving the equations”. They go on to quote a

1929 textbook recommending this approach (Hawkes, Luby, Touton, 1929). Modern textbooks also

recommend this approach, and typically present story problems as “challenge problems” and

“applications” in the back of problem-solving sections (Nathan et al., 2002). Thus, a plausible source for

teachers’ belief in the difficulty of story problems over equations is the idea that equations are needed to

solve story problems. An algebra teacher performing the problem-difficulty ranking task described in

Nathan & Koedinger (2000a) made the following reference to the translate-and-solve strategy (the

numbers 1-6 refer to sample problems teachers were given which were the same as those in Table 1):

“1 [the arithmetic equation] would be a very familiar problem… Same for 4 [the algebra equation]

… 3 [the arithmetic story] and 6 [the algebra story]) add context … Students would probably write

1 or 4 [equations] from any of the others before proceeding.”

Story Problems Can be Easier

In contrast to the common belief that story problems are more difficult than matched equations, some

studies have identified circumstances where story problems are easier to solve than equations. Carraher,

Carraher, & Schliemann (1987) found that Brazilian third graders were much more successful solving

story problems (e.g., "Each pencil costs $.03. I want 40 pencils. How much do I have to pay?") than

solving matched problems presented symbolically (e.g., "3 x 40"). Baranes, Perry & Stigler (1989) used

the same materials with US third graders. US children had higher overall success than the Brazilian

children and, unlike the Brazilian children, did not perform better in general on story problems than

symbolic problems. However, Baranes and colleagues (1989) demonstrated specific conditions under

which the US children did perform better on story problems than symbolic ones, namely, money


contexts and numbers involving multiples of 25, corresponding to the familiar value of a quarter of a

dollar.

If story problems are sometimes easier as the Carraher and colleagues (1987) and Baranes and

colleagues (1989) results suggest, what is it about the story problem representation that can enhance

student performance? Baranes and colleagues (1989) hypothesized that the situational context of story

problems can make them easier than equivalent symbolic problems. In particular, they suggested that

the problem situation activates real-world knowledge (“culturally constituted systems of quantification”,

p. 316) that aids students in arriving at a correct solution.

Such an advantage of stories over symbolic forms can be explained within the solution phase of the

problem-solving framework presented in Figure 1. Story problems can be easier when stories elicit

different, more effective, solution strategies than those elicited by equations. Past studies have

demonstrated that different strategies can be elicited even by small variations in phrasing of the same

story. For example, Hudson (1983) found that nursery school children were 17% correct on a standard

story phrasing “There are 5 birds and 3 worms. How many more birds are there than worms?”

However, performance increased to an impressive 83% when the story is phrased as “There are 5 birds

and 3 worms. How many birds don’t get a worm?” The latter phrasing elicits a match-and-count

strategy that is more accessible for novice learners than the more sophisticated subtraction strategy

elicited by the former, more standard phrasing.

The notion of a “situation model” (Kintsch & Greeno, 1985; Nathan et al., 1992) provides a

theoretical account of how story problems described in one way can elicit different strategies than

equations or story problems described in other ways. In this account, problem solvers comprehend the

text of a story problem by constructing a model-based representation of actors and actions in the story.

Differences in the stories tend to produce differences in the situation models, which in turn can

influence the selection and execution of alternative solution strategies. By this account, it is the

differences in these strategies, at the solution phase (see Figure 1) that ultimately accounts for


differences in performance. For instance, Nunes, Schliemann, & Carraher (1993) found that everyday

problems were more likely to evoke oral solution strategies whereas symbolic problems evoked less

effective written arithmetic strategies.

Developers of process models of story problem solving (e.g., Bobrow, 1968; Cummins et al., 1988;

Mayer, 1982b) have been careful to differentiate comprehension versus solution components of story

problem solving. However, readers of the literature might be left with the impression that equation

solving involves only a solution phase; in other words, that comprehension is not necessary. Although it

may be tempting to think of “comprehension” as restricted to the processing of natural language, clearly

other external forms, like equations, charts, and diagrams (cf., Larkin & Simon, 1987), must be

understood or “comprehended” to be used effectively to facilitate reasoning. The lack of research on

student comprehension of number sentences or equations may result from a belief that such processing

is transparent or trivial for problem solvers at the algebra level. Regarding equations like “(81.90 - 66)/6

= x” and “x * 6 + 66 = 81.90” in Table 2, an algebra teacher commented that these could be solved

“without thinking”.

Hypotheses and Experimental Design

The two studies presented here are the first we know of that test the common belief that algebra learners

have greater difficulty with story problems than matched equations1. Table 1 shows examples of the

main factors manipulated in these studies. In addition to the main contrast between story problems and

equations (first and last columns in Table 1), we added an intermediate problem representation we refer

to as “word problems” or “word equations” (middle column in Table 1). We included word equations

to isolate effects of situational knowledge from effects of differences in “language” comprehension

demands between verbal and symbolic forms. If stories cue useful situational knowledge, then we

should find students making fewer errors on story problems than both word and symbolic equations. If

1 Recent reviews of mathematics learning research (Kilpatrick, Swafford, Findell, 2001), story problem research

(Reed, 1998) and algebra research (Bednarz, Kieran, Lee, 1996; Kieran, 1992) do not reference any such studies.


students’ relevant symbolic comprehension skills are lagging behind their relevant verbal

comprehension skills, then we should find them making more errors on symbolic equations than either

of the verbal forms.

----- Insert Table 1 about here -----

Our goal was to explore representation effects for students at the transition from arithmetic to

algebraic competence - a domain referred to as “early algebra” (cf., Carpenter & Levi, 1999; Kaput, in

press). Thus, we included both relatively complex arithmetic problems (first row in Table 1) and

relatively simple algebra problems (second row in Table 1). In the arithmetic problems, the problem

unknown is the result of the process or sequence of operators described. These “result-unknown”

problems are more complex than those used in prior research on elementary arithmetic problem solving

(Briars & Larkin, 1984; Carpenter & Moser, 1984; Hiebert, 1982; Riley & Greeno, 1988). They

involve two arithmetic operators (e.g., multiplication and addition) rather than one, decimals rather than

just whole numbers, and more advanced symbolic notation like parentheses and equations with a

variable on the right side (e.g., “(81.90 – 66)/6 = x”). The relatively simple algebra problems we used

are two operator “start-unknowns”, that is problems where the unknown is at the start of the arithmetic

process described.

These problems are not meant to be representative of all kinds of algebraic thinking or even all

kinds of algebraic problem solving. As others have well noted (cf., Carpenter & Levi, 1999; Kaput, in

press), algebraic thinking involves more than problem solving and includes study of functional relations,

covariation, graph comprehension, mathematical modeling and symbolization, and pattern

generalization. Even within the smaller area of algebraic problem solving, there is a large variety of

problems. Others have explored more difficult classes of algebra problems than the ones used, for

instance, problems where the unknown is referenced more than once and possibly on both sides of the

equal sign, like “5.8x – 25 = 5.5x” (e.g., Bednarz & Janvier, 1996; Koedinger, Alibali, & Nathan,


submitted; Kieran, 1992). Still others have begun to explore simpler classes of problems that may draw

out algebraic thinking in elementary students (e.g., Carpenter & Levi, 1999; Carraher, Schliemann, &

Brizuela, 2000).


Our review of the literature on story problem solving leads us to consider three competing

hypotheses regarding the effects of problem representation on quantitative problem solving at the early

algebra level. The “symbolic facilitation” hypothesis is consistent with the common belief that

equations should be easier than matched story problems because difficult English language

comprehension demands are avoided (Cummins et al., 1988) and because the solution step of translating

the problem statement to an equation is eliminated (Paige & Simon, 1966). The “situation

facilitation” hypothesis follows from the idea that story representations can cue knowledge that

facilitates effective strategy selection and execution (cf., Baranes et al., 1989; Carraher et al., 1987).

Note that the four cover stories we used, shown in Table 2, all involve money quantities and familiar

contexts for junior-high aged students. The situation facilitation hypothesis predicts that students will

make fewer errors on story problems than on situation-free word-equations and equations.

A third “verbal facilitation” hypothesis follows from the observation that algebra equations are not

transparently understood and the equation comprehension skills of beginning algebra students may lag

behind their existing English language comprehension skills. The verbal facilitation hypothesis predicts

that students will make fewer errors on story and word equations than on the more abstract equations.

The verbal facilitation claim is similar to the situational facilitation and the culturally-bound results of

Carraher et al. (1987) and Baranes et al. (1989) that in some cases situational knowledge helps

elementary students perform better on story problems than matched equations. For algebra students,

however, it is not just situation knowledge that can make story problems easier. Rather, it is also that

the increased demands on symbolic comprehension presented by equations lead to difficulties for


algebra students who have by now mostly mastered the English comprehension knowledge needed for

matched verbally-stated problems. The verbal facilitation hypothesis uniquely predicts that both word

equations and story problems can be easier than matched equations2.

In contrast with the idea that equations are transparently understood, a key point of this article is to

test the claim that comprehension of algebra equations is not trivial for algebra learners and that, as a

consequence, algebra story problems can sometimes be easier to solve than matched equations. The

verbal facilitation claim directly contradicts the documented views of many teachers and education

researchers (Nathan & Koedinger, 2000a, 2000b) and cognitive scientists (e.g., Cummins et al., 1988;

Geary, 1994) that algebra story problems are inherently more difficult than matched equations. This

claim also addresses a gap in past theoretical analyses of quantitative reasoning that have provided

models of story comprehension (e.g., Bobrow, 1968; Cummins et al., 1988; Kintsch & Greeno, 1985;

Nathan, Kintsch, & Young, 1992) but have neglected equation comprehension. These models did not

address the possibility that the equation format may present greater comprehension challenges for

learners than analogous verbal forms (cf. Kirshner, 1989; Sleeman, 1984).

DIFFICULTY FACTORS ASSESSMENTS: STUDIES 1 AND 2

We investigated the symbolic, situation, and verbal facilitation hypotheses in two studies. In both

studies, students were asked to solve problems selected from a multi-dimensional space of problems

systematically generated by crossing factors expected to influence the degree of difficulty of problems.

In addition to the problem representation and unknown position factors shown in Table 1, we also

manipulated the type of the numbers involved (whole number vs. positive decimals) and the final

arithmetic that problems required (multiplication and addition vs. subtraction and division). We refer to

2 Combinations of the symbolic, situation, and verbal facilitation hypotheses are possible. For instance, combining

symbolic and situation facilitation predicts that story problems and equations will both be easier than word equations.

Alternatively, combining situation and verbal facilitation predicts that story problems will be easier than word equations

and, in turn, word equations will be easier than equations.


the methodology we employ as “Difficulty Factors Assessment” or DFA (Heffernan & Koedinger,

1998; Koedinger & MacLaren, 2002; Koedinger & Tabachneck, 1993; Verzoni & Koedinger, 1997) and

the two studies described here as DFA1 and DFA2.

Methods for DFA1 and DFA2

Subjects

The subjects in DFA1 were 76 students from an urban high school. Of these students, 58 were enrolled

in one of 3 mainstream Algebra 1 classes, and 18, who took Algebra 1 in 8th grade, were 9th graders

enrolled in a Geometry class. Four different teachers taught the classes. The subjects in DFA2 were

171 students sampled from 24 classrooms at three urban high schools. All students were in a first year

Algebra 1 course. Twelve different teachers taught the 24 classroom sections.

Form design

Ninety-six problems were created using four different cover stories that systematically vary four

difficulty factors: 3 levels of problem presentation (Story, word equation, and symbol equation) X 2

levels of unknown positions (result vs. start) X 2 number types (whole vs. decimal) X 2 final arithmetic

types (multiplication and addition vs. subtraction and division). Table 3 shows the "cross table" for

these factors; each cell is a different problem. Problems in the same column all have a common

underlying mathematical structure as indicated in the "base equation" row, for instance, all integer

Donut problems (fourth column of Table 3) are based on the equation: "4 * 25 + 10 = 110". The answer

for each problem is one or other of these bold numbers. When the final-arithmetic is multiply-add (see

“*, +” in the third column), the answer is the last number (e.g., 110) and when it is subtract-divide

(“-, /”), the answer is the first number (e.g., 4). The unknown-position (second column), as described

above, determines whether the unknown is the result or start of the arithmetic operations described in

the problem. Note that the initial appearance of the result-unknown, subtract-divide problems (e.g.,

“(110 – 10) / 25 = x”) and start-unknown, multiply-add problems (e.g., “(x – 10) / 25 = 4”) is reverse

that shown in the base equation form.



The 96 problems were distributed onto sixteen forms with eight problems on each form. We

limited the number of items per form because prior testing revealed that eight problems could be

reasonably completed in less than half a class period (18 minutes). This decision reflected teachers’

recommendations for reasonable quiz duration and for use of class time relative to the demands of the

required curriculum.

Placement of problems on forms met the following constraints. Each form included two story

result-unknowns, two story start-unknowns, one word equation result-unknown, one word equation start

unknown, one equation result unknown and one equation start unknown. Each form had only one

problem from a particular base equation (one from each column in Table 3) such that there was no

repetition of the numbers or answers on any form. The four story problems on every form had either all

integer number types or all decimal number types. The four other problems had the opposite number

types. The forms were also designed to counter-balance problem order, described below, as prior

testing revealed that student performance tended to decline on problems near the end of the test.

Each form had twice as many story problems (4) as word equations (2) or symbol equations (2).

The 32 story problems were used twice, once on eight forms in which the story problems came first and

again on eight "reversed" forms in which the story problems came last (in reverse order). The order of

the word equations and equations were also counter-balanced. The word equation and equation

problems on the reversed forms are identical in their difficulty factor composition to their counter-parts

on the normal forms (i.e., each such pair has the same unknown position, number type, and final

arithmetic), but differ in the base equation.

The form design of DFA1 and DFA2 was identical with one exception. The base equation for the

decimal Donut problems was changed on DFA2 from 7 * .37 + .22 = 2.81 to .37 * 7 + .22 = 2.81 so

that, like all the other decimal base equations, the answer to the unknown is always a decimal (e.g., .37

or 2.81) and not a whole number (e.g., 7).


Procedure

The quiz forms were given in class during a test day. Students were given 18 minutes to work on the

quiz and were instructed to show their work, put a box around their final answer, and not to use

calculators. We requested that students not use calculators so that we could better see students’ thinking

process in the arithmetic steps they wrote down. The procedures for DFA1 and DFA2 were identical.

Results for DFA1 and DFA2

DFA1 Results

We performed both an item analysis and subject analysis to assess whether the results generalize across

both the item and student populations (cf., Clark, 1972). For the item analysis, we performed a three

factor ANOVA with items as the random effect and the three difficulty factors: representation, unknown

position, and number type as the fixed effects3.

We found evidence for main effects of all three factors. Students (n=76) performed better on

story problems (66%) and word equations (62%) than equations (43%; F(2, 108) = 11.5, p < .001);

better on result-unknown problems (66%) than on start-unknown problems (52%; F(1, 108) = 10.7, p

< .002); better on whole number problems (72%) than decimal number problems (46%; F(1, 108) = 44,

p < .001). A Scheffe's S post-hoc showed a significant difference between story problems and equations

(p < .001), word equations and equations (p < .01), but not story problems and word equations (p = .80).

None of the interactions were statistically significant.

These results contradict the symbolic facilitation hypothesis because story problems are not

harder than equations. They contradict the situation facilitation hypothesis because story problems are

not easier than word equations. The results support the verbal facilitation hypothesis because word

3 Because the dependent variable is a proportion (students solving the item correctly divided by the number of students

who saw the item) we used a logit transformation as recommended by Cohen & Cohen (1973, pp. 254-259): .5 * ln(p / (1-

p)) where p=0 is replaced by p=1/(2N) and p=1 by p=1-1/(2N) and N is number of students who see the item. Note,

analyses using the proportions without transformation yield quite similar results and do not move any p-values across the

0.05 threshold.


equations are substantially easier than equations. In other words, it is the difference between verbal and

symbolic representation not the difference between situational context and abstract description that

accounts for the observed performance differences.

Figure 2 illustrates the main effects of the three difficulty factors: representation, unknown position,

and number type. As is evident, the effect of representation is primarily a consequence of the difference

between the symbolic equation representation and the two forms of verbal representation, story and

word equations. It appears that there may be a difference between story and word equation

representations for the decimal problems (see the graph on the right in Figure 2), however, the

interaction between representation and number type is not statistically significant (F(2, 108) = .82, p

= .44).


To confirm that the main effects of the three difficulty factors generalize not only across items but

also across students (cf., Clark, 1972), we performed three separate one factor repeated measures

ANOVAs for each of the three difficulty factors: representation, unknown position, and number type.

Unlike the item analysis above, the random effect here is students rather than items. Consistent with the

item analysis, the repeated measures student analyses revealed significant main effects of each factor.

Story and word problems are significantly easier than equations (F(2, 150) = 8.35, p < .001). Result-

unknowns are significantly easier than start-unknowns (F(1, 75) = 19.8, p < .001). Whole number

problems are significantly easier than decimal number problems (F(1,75) = 42.3, p < .001).

DFA2 Results

The results of DFA2 replicate the major findings of DFA1 for the main effects of representation,

unknown position and number type (see Figure 3). Students (n=117) were 67% correct on the whole

number problems and only 54% correct on the decimal problems (F(1, 116) = 22, p < .001). They were

more often correct on result-unknown problems than start-unknown problems (72% vs. 49%; F(1, 116)


= 49, p < .001). And, as in DFA1, students performed best on story problems (70%), next best on word

equations (61%) and substantially worse on equations (42%; F(2, 116) = 22, p < .001). Again, the big

difference is between the word equations and equations (p < .001 by a post-hoc Scheffe's S test). Like

DFA1, the difference between story problems and word equations is not statistically significant (p = .09

by a post-hoc Scheffe's S test). However unlike DFA1, there is a significant interaction between

representation and number type (F(2, 116) = 3.7, p = .03) as illustrated in Figure 3. When the number

type is whole number, there is no difference between story and word equations (72% vs. 73%). Only

when the number type is decimal does an advantage for story problems over word equations appear

(68% vs. 48%, p < 01 by a post-hoc contrast test).


As can be seen in Figure 2, this interaction was apparent in DFA1 although it was not statistically

significant. However, to further evaluate the situation facilitation hypothesis, we tested whether the

advantage of story over word on decimal problems generalizes across the two studies. The difference

between the decimal story problems and decimal word equations is statistically reliable when combining

the data from the two studies (64% vs. 47%, p < .01 by a post-hoc Scheffe's S test including just the

decimal items). This result leads us to conclude that situation facilitation does exist, but its scope is

limited. In these data, problem-solving performance is only facilitated by the richer story contexts,

above and beyond the more general verbal advantage, when decimal numbers are used. We propose an

explanation for this effect when we report findings of the strategy and error analyses in the next section.

As in the student analysis in DFA1, a one factor repeated measure ANOVA was conducted for each

of the three difficulty factors with student as the random factor. The generalization of these effects

across students was confirmed. Representation, unknown position, and number difficulty all had

statistically significant effects: F(2, 340) = 37, p <.001, F(1, 170) = 137, p < .001, and F(1,170) = 20, p

< .001, respectively.


STRATEGY AND ERROR ANALYSES OF LEARNERS

Toward the goal of providing a cognitive process explanation of our results, we performed a detailed

strategy and error analysis of student solution traces. This method serves as a qualitative analysis of

students' written protocols. This analysis and the coding categories used were strongly influenced by

cognitive modeling work within the ACT-R theory and production system (Anderson, 1993). The

details of our ACT-R model of early algebra problem solving are beyond the scope of the current paper.

We refer the interested reader to Koedinger & MacLaren (2002).

Qualitative Strategy Analysis – Stories Can be Solved Without Equations

The results of DFA1 and DFA2 contradict the common belief in the ubiquitous difficulty of story

problems as well as predictions of teachers and education researchers (Nathan & Koedinger, 2000a),

and cognitive researchers (Cummins et al., 1988). A common argument supporting this belief is that

story problems are more difficult than matched equations because students must translate the story into

an equation in order to solve it (Bobrow, 1968; Hawkes, Luby, Touton, 1929; Paige & Simon, 1966).

Indeed, we did observe examples of this normative translation strategy in student solutions (see Figure

4a). However, we found many students using a variety of informal strategies as well (c.f., Hall, Kibler,

Wenger, and Truxaw, 1989; Katz, Friedman, Bennett, & Berger, 1996; Kieran, 1992; Resnick, 1987;

Tabachneck, Koedinger & Nathan, 1994). By informal we mean that students do not rely on the use of

mathematical (symbolic) formalisms, like equations. We also mean that these strategies and

representations are not acquired typically through formal classroom instruction. Figures 4b-d show

examples of these informal strategies as observed in students' written solutions.

Figure 4b shows the application of the guess-and-test strategy to a start-unknown word problem. In

this strategy, students guess at the unknown value and then follow the arithmetic operators as described

in the problem. They compare the outcome with the desired result from the problem statement and if

different, try again. In the solution trace shown in Figure 4b, we do not know for sure where the student

started writing -- perhaps with the scribbled out work on right -- but it is plausible that her first guess at


“some number” was 2. Just to the right and somewhat below the question mark at the end of the

problem, we see her written arithmetic applying the given operations multiply by .37 and add .22 to her

guess of 2. The result is .96, which she sees is different than the desired result of 2.81and perhaps also,

that it is substantially lower than 2.81. It appears her next guess is 5, which yields a result (2.07) that is

closer but still too small. She abandons a guess of 6, perhaps because she realizes that 2.07 is more than

.37 short of 2.81. She tries 7 which correctly yields 2.81 and we see the student writes “The number is

7”. The guess-and-test strategy is not special to early algebra students, but has also been observed in the

algebraic problem solving of college students (e.g., Hall, Kibler, Wenger, and Truxaw, 1989; Katz,

Friedman, Bennett, & Berger, 1996; Tabachneck, Koedinger & Nathan, 1994).

Figure 4c illustrates a second informal strategy for early algebra problem solving we call the unwind

strategy. To find the unknown start value, the student reverses the process described in the problem.

The student addresses the last operation first and inverts each operation to work backward to obtain the

start value. In Figure 4c, the problem describes two arithmetic operations, subtract 64 and divide by 3,

in that order. The student starts (on the right) with the result value of 26.50 and multiplies it by 3 as

this inverts the division by 3 that was used to get to this value. Next the student takes the intermediate

result, 79.50, and adds 64.00 to it as this inverts the subtraction by 64 described in the problem. This

addition yields the unknown start value of 143.50.

Although the informal guess-and-test strategy appears relatively inefficient compared to the formal

translation strategy (see the amount of writing in Figure 4b compared with 4a), the informal unwind

strategy actually results in less written work than the translation strategy (compare 4c and 4a). In

unwind, students go directly to the column arithmetic operations (see 4c) that also appear in translation

solutions (see the column subtraction and division in 4a), but they do so mentally and save the effort of

writing equations. (For other examples of such “mental algebra” amongst more expert problem solvers

see Hall et al., 1989; Tabachneck, Koedinger & Nathan, 1994). The next section addresses whether

students use informal strategies frequently and effectively.


Quantitative Strategy Analysis – Words Elicit More Effective Strategies

We coded student solutions for the strategies apparent in their written solutions for DFA1 and DFA2.

Our strategy analysis focuses on the early algebra start-unknown problems (shown in Figure 4a-d). We

observed little variability in students’ strategies on the result-unknown problems. Although some

students translated verbal result-unknown problems into equations (see Figure 4e), in the majority of

solutions, students went directly to the arithmetic. Typical solution traces included only arithmetic

work as illustrated in the lower left corner of Figure 4e (i.e., all but the three equations).

Table 4 shows the proportion of strategy use on start-unknown problems for the three different

representations. Different representations elicited different patterns of strategy usage. Story problems

elicited the unwind strategy most often, 50% of the time. Story problems seldom elicited the symbolic

translation strategy typically associated with algebra (only 5% of the time). Situation-less word

equations tended to elicit either the guess-and-test (23% of the time) or unwind (22%) strategy.

Equations resulted in no response 32% of the time, more than twice as often as the other representations.

When students did respond, they tended to stay within the mathematical formalism and apply symbol

manipulation methods (22%). Interestingly even on equations, students used the informal guess-and-

test (14%) and unwind (13%) strategies fairly often. In fact, as Figure 4d illustrates, sometimes students

translate a verbal problem to an equation but then solve the equation sub-problem informally, in this

case, using the unwind strategy.

Story problems may elicit more use of the unwind strategy than word equations perhaps because of

their more episodic or situated nature (cf., Hall et al., 1989). Retrieving real world knowledge, for

example, about a box of donuts (see Figure 1) may support students in making the whole-part inference

(cf., Greeno, 1983; Koedinger & Anderson, 1990) that subtracting the box cost from the total cost gets

one closer to the solution.

We also saw that word equations elicit more unwind strategy usage than equations. Although word

equations are situation-less, we did use words like " starting with", "and then", and "I get" that describe


someone performing an active procedure. Even the mathematical operators were described as actions,

"multiply" and "add", instead of relations, "times" and "plus". Perhaps this active description makes it

easier for students to think of reversing the performance of the procedure described. We suggest future

research that tests this “action facilitation” hypothesis, and contrasts our current "procedural" word

equations with "relational" word equations like "some number times .37 plus .22 equals 2.81" (see

Figure 1 for the analogous procedural word equation). Action facilitation predicts better student

performance on procedural than relational word equations whereas verbal facilitation predicts equal

performance (with both better than equations).

In addition to investigating differences in strategy selection, we also analyzed the effectiveness of

these strategies. Table 5 shows effectiveness statistics (percent correct) for the unwind, guess-and-test,

and symbol manipulation strategies on start-unknown problems in all three representations. The

informal strategies, unwind and guess-and-test, showed a higher likelihood of success (69% and 71%

respectively) than use of the symbol manipulation approach (51%). So, it appears one reason these

algebra students did better on story and word problems than equations is they select more effective

strategies more often. However, this effect could result from students choosing informal strategies on

easier problems. Use of a no-choice strategy selection paradigm (Siegler & Lemaire, 1997) is a better

way to test the efficacy of these strategies. Nhouyvanisvong (1999) used this approach to compare

equation solving and guess-and-test performance on story problems normatively solved using a system

of two equations and/or inequalities. Surprisingly, he found students instructed to use guess-and-test

were more successful than those instructed to use equation solving.

Error Analysis – Comprehending Equations is Harder than Comprehending Words

Our analysis of student strategies and, in particular, the differential use of informal strategies provides

one reason why story and word problems can be easier than matched equations. Our analysis of student

errors provides a second reason. Unlike the first and second grade students in Cummins et al. (1988),

who have yet to acquire critical English language comprehension skills, high school algebra students


have more developed English comprehension skills, but are still struggling to acquire critical Algebra

language comprehension skills.

A categorization of student errors into three broad categories -- 1) no response, 2) arithmetic error,

and 3) other conceptual errors -- provides insight into how students process story problems and

equations differently. Students’ solutions were coded as no response if nothing was written down for

that problem. Systematic occurrences of no response errors suggest student difficulties in

comprehending the external problem representation. Our counter-balancing for the order of problem

presentation allows us to rule out student fatigue or time constraints. Student solutions were coded as

arithmetic error if a mistake was made in performing an arithmetic operation, but the solution was

otherwise correct (e.g., Figure 5f). Arithmetic errors indicate correct comprehension and, in the case of

start-unknowns, correct formal or informal algebraic reasoning. Apart from some rare (19 errors out of

1976 solutions) non-arithmetic slips, like incorrectly copying a digit from the problem statement, all

other errors were coded as conceptual errors. Examples of a variety of different conceptual errors are

shown in Figures 5a-5e.

Figure 6a shows the proportions of the error types for the three levels of the representation factor:

story, word-equation, or equation. The key difference between equations and the two verbal

representations (story, word-equation) is accounted for primarily by no response errors (26% = 127/492

vs. 8% = 119/1465). No response errors imply difficulty comprehending the external problem

representation. These data suggest that students in these samples were particularly challenged by the

demands of comprehending the symbolic algebra representation. The language of symbolic algebra

presents some new demands that are not common in English or in the simpler symbolic arithmetic

language of students’ past experience (e.g., one operator number sentences, like “6 - 2 = ?”, used in

Cummins et al., 1988). The algebraic language adds new lexical items, like “x”, “*”, “/”, “(“, and new

syntactic and semantic rules, like identifying sides of an equation, interpreting the equals sign as a

relation rather than an operation, order of arguments, and order of operators. When faced with an


equation to solve, students lacking aspects of algebraic comprehension knowledge may give up before

writing anything down.

Further evidence of students’ difficulties with the “foreign language of algebra” comes from

students’ conceptual errors. As shown in Figure 5a, students make more conceptual errors on equations

(28% = 103/365) than on word equations (23% = 103/450) and particularly story problems (16% =

144/896)4. Figures 6a and 6b show examples of conceptual errors on equations. In Figure 6a we see an

order of operations error whereby the student performs the addition on the left-hand side (.37 + .22)

violating the operator precedence rule that multiplication should precede addition. We found a

substantial proportion of order of operation errors on equations (4.9% = 24/492), while order of

operations errors on verbal problems were extremely rare (0.2% = 4 /1468).

In Figure 6b we see two examples of algebra manipulation errors. This student appears to have

some partial knowledge of equation solving, namely, that you need to get rid of numbers by performing

the same operation to “both sides”. The student, however, operates on both sides of the plus sign rather

than both sides of the equal sign. These errors indicate a lack of comprehension of the quantitative

structure expressed in the given equations. Such errors appear less frequently on word and story

problems indicating that comprehension of the quantitative structure is easier for students when that

structure is expressed in English words rather than algebraic symbols.

Explaining Situation Facilitation

Algebraic language acquisition difficulties, such as comprehension and conceptualizing the underlying

quantitative relations, account for much of the error difference between equations and verbal problems.

These differences are consistent with the verbal facilitation hypothesis. However, we also observed a

smaller situation facilitation effect whereby story performance was better than word-equation

performance under certain conditions – namely when dealing with decimal numbers. This interaction

was statistically reliable in DFA2, but the same trend was also apparent in the smaller DFA1 data set. 4 The proportion of conceptual errors reported here is conditional on there having been a response -- solutions with no

response errors are not counted in the denominator.


Figure 6b shows the proportions of the three error types for the representation and number type factors

together. The interaction between representation and number type is caused largely by fewer arithmetic

errors on decimal story problems (12% = 46/389 on decimal vs. 5% = 17/363 on whole) than for word-

equations or equations (23% = 33/144 on decimal vs. 2% = 4/203 on whole)5. A common error on

situation-less word and equation problems was to miss-align place values in decimal arithmetic (see

Figure 5f). In contrast this error was rare on story problems. It appears the money context of the story

problems helped students to correctly add (or subtract) dollars to dollars and cents to cents. In contrast,

without the situational context (in word and equations), students would sometimes, in effect, add dollars

to cents.

Situation-induced strategy differences also appear to contribute to students’ somewhat better

performance on story problems than word equations. As we saw from the strategy analysis, students

were more likely to use the unwind strategy on story problems (50%) than on word-equations (22%) or

equations (13%). The unwind strategy may be less susceptible to conceptual errors than the guess-and-

test strategy which was used more frequently on word-equations. A particular weakness of the guess-

and-test strategy is the need to iterate through guesses until a value is found that satisfies the problem

constraints. As illustrated in Figure 6e, a common conceptual error in applying guess-and-test is to give

up before a satisfactory value is found. Guess-and-test is more difficult when the answer is a decimal

rather than a whole number because it takes more iterations in general to converge on the solution. This

weakness of guess-and-test and its greater relative use on word equations than story problems may

account for the greater number of conceptual errors on decimal word-equations than decimal story

problems shown in Figure 6b.

5 The proportion of arithmetic errors reported here is conditional on the solution being conceptually correct -- solutions

with no response or conceptual errors are not counted in the denominator.


DISCUSSION

Assessing the Symbolic, Situation, and Verbal Facilitation Hypotheses

In the introduction we contrasted three hypotheses regarding the effects of different problem

representations on algebra problem solving. The symbolic facilitation hypothesis predicts that story

problems are more difficult than matched equations because equations are more parsimonious and their

comprehension more transparent. Our results with high school students solving entry-level algebra

problems in two different samples contradict this claim and show, instead, that symbolic problems can

be more difficult for students, even after a year or more of formal algebra instruction.

Alternatively, the situation facilitation hypothesis follows from situated cognition research (Baranes

et al., 1989; Brown, Collins, & Duguid, 1989; Carraher et al., 1987; Nunes, Schliemann, & Carraher,

1993) and suggests that problem situations facilitate student problem solving because they

contextualized the quantitative relations. The prediction of the situation facilitation hypothesis, that

story problems are easier than both word-equations and equations, is not fully consistent with our

results. While students in DFA1 and DFA2 did perform better on story problems than equations, they

also performed better on word-equations than equations. Thus, it is not simply the situated nature of

story contexts that accounts for better performance.

The verbal facilitation hypothesis focuses not on the situated nature of story problems per se, but on

their representation in familiar natural language. The hypothesis follows from the idea that students,

even after an algebra course, have had greater experience with verbal descriptions of quantitative

constraints than with algebraic descriptions of quantitative constraints. Thus, it predicts that word-

equations, as well as story problems, will be easier than mathematically equivalent equations. The

prediction relies on two claims. First, students initially have more reliable comprehension knowledge

for verbal representations than symbolical ones. Second, verbal representations better cue students’

existing understanding of quantitative constraints and, in turn, informal strategies or weak-methods for


constraint satisfaction, like generate-and-test or working backwards. Early algebra students appear

better able to successfully use such strategies than the symbol-mediated equation solving strategy.

This second claim, that verbal representations help cue students’ knowledge, is consistent with the

assertion made by Nunes, Schliemann, & Carraher (1993, p. 45) that "discrepant performances can be

explained in terms of the symbolic systems being used." They found evidence that the chosen symbolic

system (e.g., formal vs. verbal/oral) determines performance more than the given one. Students

performed better on all kinds of problems, whether abstract or situational, when they used informal oral

strategies than when they used formal written strategies. Our strategy analysis revealed analogous

results for older students and a different class of problems. In addition to providing further evidence to

support the explanation provided by Nunes, Schliemann, & Carraher (1993), our results extend that

explanation. In our error analysis, we found evidence that the given representation has direct effects on

student performance beyond the indirect effects it has on influencing student strategy choice. More

students failed to comprehend given equation representations than given verbal representations as

indicated by a greater frequency of no-response and conceptual errors on the former (as illustrated in

Figure 6a).

Like Baranes et al. (1989), we did see some localized situational facilitation in students’

performance on story problems. First, students used the unwind strategy more often on story problems

than on word-equations or symbolic equations. Because this strategy is more reliable than equation

manipulation and more efficient than guess-and-test, students were less likely to make conceptual errors

when solving decimal story problems. A second situational effect involves support for decimal

alignment in the context of a story problem. Students avoided adding dollars to cents in the story

context and thus made fewer arithmetic errors on problems involving decimals in this context than in

content-free word equations and equations.

One might interpret some educational innovations emphasizing story contexts (e.g., CTGV, 1997;

Koedinger, Anderson, Hadley, & Mark, 1997), calls for mathematics reform (e.g., NCTM, 2000), and


situated cognition and ethnomathematical research (e.g., Brown, Collins, & Duguid, 1989; CTGV,

1990; Greeno & MMAP Group, 1998; Roth, 1996) as suggesting that “authentic” problem situations

generally help students make sense of mathematics. In contrast, our results are consistent with those of

Baranes et al. (1989) that situational effects are specific and knowledge related. Similarly, Nunes,

Schliemann, & Carraher (1993, p. 47) argued that the differences they observed "cannot be explained

only by social-interactional factors". Indeed, as long as problems were presented in a story context, they

found no significant difference between performances in different social interaction settings, whether a

customer-vendor street interaction or a teacher-pupil school interaction.

Situational effects are not panaceas for students’ mathematical understanding and learning. Clearly,

though, problem representations, including their embedding and referent situations, have significant

effects on how students think and learn. Better understanding of these specific effects should yield

better instruction.

Two Reasons Why Story and Word Problems Can Be Easier

Two key reasons explain the surprising difficulty of symbolic equations relative to both word and story

problems. These two reasons correspond, respectively, to the solution and comprehension phases of

problem solving illustrated in Figure 1. First, students’ access to informal strategies for solving early

algebra problems provides an alternative to the logic that word problems must be more difficult because

equations are needed to solve them. Our data as well as that of others (cf., Stern, 1997; Hall et al., 1989)

demonstrates that solvers do not always use equations to solve story problems. Second, despite the

apparent ease of solving symbolic expressions for experienced mathematicians, the successful

manipulation of symbols requires extensive symbolic comprehension skills. These skills are acquired

over time through substantial learning. Early in the learning process, symbolic sentences are like a

foreign language – students must acquire the implicit processing knowledge of equation syntax and

semantics. Early algebra students’ weak symbolic comprehension skills are in contrast to their existing

skills for comprehending and manipulating quantitative constraints written in English. When problems


are presented in a language students understand, students can draw on prior knowledge and intuitive

strategies to analyze and solve these problems despite lacking strong knowledge of formal solution

procedures.

One dramatic characterization of our results is that under certain circumstances students can do as

well on simple algebra problems as they do on arithmetic problems. This occurs when the algebra

problems are presented verbally (59% and 60% correct on start-unknown stories in DFA1 and DFA2)

and the arithmetic problems are presented symbolically (51% and 56% correct on result-unknown

equations in DFA1 and DFA2). While we have been critical of the sweeping claim made by Cummins

and colleagues (1988) that “students … continue to find word problems … more difficult to solve than

problems presented in symbolic format (e.g., algebraic equations)", we agree on the importance of

linguistic development. But, we broaden the notion of linguistic forms to include mathematical symbol

sentences. Cummins emphasized that the difficulty in story problem solving is not specific to the

solution process, but is also in the comprehension process. “The linguistic development view holds that

certain word problems are difficult to solve because they employ linguistic forms that do not readily

map onto children’s existing conceptual knowledge structures.” (Cummins, et al, 1988, p. 407). A

main point from our results is that the difficulty in equation solving is similarly not just found in the

solution process, but as much or more so in the comprehension process. Algebra equations employ

linguistic forms that beginning algebra students have difficulty mapping onto existing conceptual

knowledge structures.

Although our results are closer to those who have found advantages for story problems over

symbolic problems under some circumstances (Baranes, Perry & Stigler, 1989; Carraher, Carraher, &

Schliemann, 1987), our analysis of the underlying processes has important differences. Like the

situated cognition researchers we did find particular circumstances where the problem situation

facilitated performance (money and decimal arithmetic). However, our design and error analysis

focused not only on when and why story problems might be easier, but also on when and why equations


might be harder. The key result here is that equations can be more difficult to comprehend than

analogous word problems, even though both forms have no situational context. Kirshner (1989) and

Sleeman (1984) have also highlighted the subtle complexities of comprehending symbolic equations and

Heffernan & Koedinger (1998) have identified similar difficulties in the production of symbolic

equations.

Logical Task-Structure vs. Experience-Based Reasons for Difficulty Differences

An important question regarding these results is whether difficulty factor differences are a logical

consequence of the task structure or a consequence of biases in experience as determined by cultural

practices (e.g., presenting students with algebraic symbolism later, as in the US, versus early, as in

Russia and Singapore). The cognitive modeling work we have done (Koedinger & MacLaren, 2002)

has led to the observation that some difficulty factors are a fixed consequence of task structure, while

others are experience based.

The greater difficulty of start-unknown problems over result-unknown problems is a logical

consequence of differences in task structure. When students solve a start-unknown problem they must

do everything they need to do to solve an otherwise equivalent result-unknown problems (e.g.,

comprehend the problem statement, perform arithmetic operations) in addition to dealing with the fact

that the arithmetic operations cannot be simply applied as described in the problem. Thus, start-

unknown problems are logically constrained to be at least as difficult as result-unknown problems.

In contrast, the difficulty difference between word problems and equations is not logically

constrained. While there is some overlap in the knowledge required to solve word problems and

equations (e.g., arithmetic and operator inversion or guess-and-test skills), each problem category

requires some knowledge that the other does not. In particular, word problems require knowledge for

verbal comprehension not needed for equations. Conversely, equations require knowledge for

comprehending symbolic notation not needed for word problems. Thus, the difficulty difference


between these problem categories is not fixed. It depends on students’ relative experience with verbal

and symbolic representations.

The proportion of a student's exposure to quantitative constraints in verbal versus symbolic form

may depend, in turn, on cultural factors. Given the prevalence of natural language for other

communicative needs, early algebra students are likely to have more reliable knowledge for

comprehending verbal descriptions than symbolic ones. Thus, students may tend to find comprehending

word problems easier than equations, at least initially. However, in an educational culture where

symbolic algebra representations are experienced earlier and more frequently by students, we would

expect the difference to shrink over time. In contrast to the approach in traditional US curricula of

introducing start-unknown "number sentences" using boxes or blanks (e.g., "__ + 5 = 8") to represent

unknowns, students in other countries (Russia, Singapore) are introduced to the use of letters (e.g., "x +

5 = 8") to represent unknowns in the elementary grades (cf., Singapore Ministry of Education, 1999). If

our study was replicated in such countries, we expect that students would not experience the kind of

equation comprehension difficulties observed here and thus may show as good or better performance on

symbolic equations relative to story problems. Perhaps future cross-cultural research can test this

prediction.

Instructional Implications

We discuss the advantages and disadvantages of four alternative instructional strategies that might

appear to follow from our results. First, to the extent that people can effectively solve problems without

symbolic equations, one might well ask, why are algebra equations and equation solving skills needed at

all? One reason is that algebraic symbolism has uses outside of problem solving, including efficiently

communicating formulas and facilitating theorem proving. But even within problem solving, equations

may well facilitate performance for more complex problem conditions (Koedinger, Alibali, & Nathan,

submitted; Verzoni & Koedinger, 1997). Replacing algebra equations with alternative representations is


an intriguing idea (cf., Cheng, 1999; Koedinger & Terao, 2002), but we do not advocate eliminating

equations and equation solving from the curriculum.

Since equation solving is so hard for students to learn and because it is important, a second

instructional strategy worth consideration is a mastery-based approach (Bloom, 1984) that focuses on

equation-solving instruction in isolation. This strategy targets the syntax of algebraic transformation

rules and does not address algebraic symbolism as a representational language with semantics.

Instruction that isolates transformational rules may reduce cognitive load and thus facilitate learning

(Sweller, 1988). On the other hand, practicing procedures without an understanding of underlying

principles often leads to fragile knowledge that does not transfer well (Judd, 1908; Katona, 1940). To

be sure, students often make errors in equation solving that clearly violate the underlying semantics

(Figure 5ab; Payne & Squibb, 1990).

A third alternative takes a developmental view and suggests starting with instructional activities

involving story problems, which are easier for students to solve, and moving later to more abstract

word-equation problems and then symbolic equations. In this view, decomposing instruction to focus on

difficult equation solving skills is fine. However, such instruction should come after students have

learned the meaning of algebraic sentences, in other words, after they have learned to translate back and

forth between English and algebra. This "progressive formalization" sequencing (Romberg & de Lange,

2002) is unlike many current textbooks that teach equation solving before analogous story problem

solving (Nathan, Long, & Alibali, 2002). Our theoretical analysis (Koedinger & MacLaren, 2002) and

experiments with simulated students (MacLaren & Koedinger, 2002) support the hypothesis that

developing students’ understanding of quantitative constraints in the verbal representation should

facilitate learning of the symbolic representation of quantitative constraints.

The fourth instructional alternative advocates that students develop some competence at encoding

and representing constraints in verbal form prior to and as the basis for equation solving instruction.

Here, students’ understanding of verbal constraints is used as a bridge for understanding and


manipulating symbolic constraints. Activities and exercises should focus on translating back and forth

between verbal and symbolic representations. According to this view, instruction should aid students in

making explicit connections between their existing verbal knowledge and the new symbolic knowledge

they are learning. In this approach, instruction aims to help students to understand the meaning of

algebraic sentences and symbol-manipulation procedures by grounding symbolic forms in students’ pre-

existing verbal comprehension and strategic competence. Relevant grounding activities might include:

1) matching equations and equivalent word equations, 2) translating equations to story problems and

solving both, 3) solving story problems and summarizing both the story and the solution in equations, 4)

making students’ aware of the algorithmic nature of their intuitive verbal strategies, and generalizing

these procedures to include symbolic formalisms. We have some indication that this approach is

effective for teaching 6th graders to solve story and word equations first informally, and then

symbolically (Nathan & Koedinger, 2000c). We have also begun to experimentally compare this

approach to other approaches in extended interventions with seventh and eight grade students (Nathan,

Stephens, Masarik, Alibali, & Koedinger, 2002).

Grounding new representations to familiar representations has the potential to promote reliable

performance by facilitating meaning making and self-monitoring processes. For example, solution steps

generated in the abstract equation representation can be checked against the results of reasoning with a

concrete story representation. Errors in use of the equation representation may be detected and fixed

through comparison with the analogous story solution. Further, steps in story solutions can be used as

examples to learn, via analogy, transformations in the equation representation. Instruction that focuses

on bridging permits use of fallback strategies that can help students to acquire and debug new more

powerful abstract knowledge.

Some prior studies indicate the promise of bridging instruction in algebra learning. Nathan,

Kintsch, and Young (1992) demonstrated that student learning of algebraic symbolization for “multiple-

unknown” problems (e.g., translating more complex story problems into equations like “12(t-2) +


8t = 76”) was enhanced by encouraging students to coordinate between a grounded representation, in

this case an animation of the problem situation, and the abstract symbolic representation. Koedinger

and Anderson (1998) demonstrated that student learning of algebra symbolization for simpler linear

start-unknowns (e.g., translating story problems into expressions like “42x + 35”) was enhanced by

having students first solve story problems and then use the emergent arithmetic procedure to essentially

induce an algebraic expression. In more applied work, a key feature of the Cognitive Tutor Algebra

curriculum and software (Koedinger, Anderson, Hadley, & Mark, 1997) is to help students learn more

abstract algebraic representations (e.g., symbols and graphs) by bridging off existing knowledge in more

grounded representations (e.g., situations, words, and tables of values). Classroom field studies have

demonstrated that Cognitive Tutor Algebra results in dramatic improvements in student achievement

relative to traditional algebra curricula (Koedinger et al., 1997). Brenner, Mayer, Moseley, Brar, Duran,

Reed, & Webb (1997) demonstrated the effectiveness of a similar approach employing multiple

representations in supporting algebra learning.

CONCLUSION

Studies in a variety of domains have shown that “different presentation formats elicit qualitatively

different solution strategies” (Mayer, 1982a, p. 448) and can lead to dramatically different performance

(Kotovsky, Hayes, & Simon, 1985; Zhang & Norman, 1994). We have identified two ways in which

external representations can change performance. First, performance may change because one external

representation is more difficult to comprehend than another. Second, performance can change because

different external representations may be comprehended in different ways and in turn cue different

processing strategies.

We have found it useful to think of the process of learning formal representations (e.g., algebra,

vectors, chemical equations, etc.) as a kind of foreign language learning. Students acquire some explicit

knowledge of the grammar of algebra, like what a “term” or a “factor” is. However, it is likely that

much of the knowledge of parsing algebraic equations is perceptual learning of “chunks” (cf. Servan-


Schreiber & Anderson, 1990) that implicitly characterize the syntactic structure of expressions and

equations (cf., Berry & Dienes, 1993; Kirshner, 1989; Reber, 1967). Students may also acquire some

explicit knowledge of semantic mappings of grammatical structures, like precedence rules for order of

operations. However, just as at the syntactic level, we believe substantial implicit knowledge is acquired

at the semantic and pragmatic levels. For instance, Alibali and Goldin-Meadow (1993) have shown that

when students are first learning to solve problems like “3 + 4 + 5 = __ + 5” they often have implicit

knowledge of alternative hypotheses for meaning of the equal sign that are independently revealed in

speech and gesture. A student solving this problem may simultaneously speak in accord with one

hypothesis (“=” means one must give an answer and say 12 or 17) yet gesture (pointing at the “5” on

both sides) in accord with another (“=” means balance).

To the extent that fluency with mathematical symbols is acquired largely through implicit learning

processes (Berry & Dienes, 1993; Reber, 1967), those who have developed such expertise cannot easily

and directly reflect on the difficulties learners’ must overcome. This observation is consistent with

Nathan & Koedinger’s (2000a, 2000b) survey results that algebra teachers and educators typically judge

equations to be easier than matched story and word problems, contrary to the results reported here. As

we gain expertise in concise symbolic languages, like equations, it feels as though we can transparently

understand and solve them, as one teacher said, “without thinking”. It is tempting to project this ease

onto students and not recognize the difficulties learners’ experience in learning to comprehend and use

mathematical symbols (e.g., Figures 5a and 5b). At the same time, our expertise and corresponding

awareness of the formal method for solving a problem (e.g., Figure 4a) can lead us to underestimate the

informal understandings and strategies (e.g., Figures 4b and 4c) that students may bring to a content

domain.

We have coined the phrase expert blind spot to refer to this tendency, on one hand, to overestimate

the ease of acquiring formal representation languages, and on the other hand, to underestimate students’

informal understandings and strategies (Koedinger & Nathan, 1997; Nathan & Koedinger, Alibali,


2001). Expert blind spot has clear consequences for the design and delivery of instruction. A teacher or

textbook writer cannot effectively provide instruction that builds on students’ prior knowledge

(Bransford, Brown, & Cocking, 1999) if they employ incorrect assumptions about what in-coming

students know, what activities are particularly difficult, and what activities elicit informal

understandings.

Methods of cognitive task analysis, like the “difficulty factors assessment” and the fine-grained

analyses of students’ strategies and errors used in these studies, can help to develop detailed theories of

learner knowledge, implicit and explicit, and how external representations and activities evoke that

knowledge. Such research is an important activity for the learning sciences because much of

educational decision making may be incorrectly biased by explicit knowledge and beliefs that are at

odds with the reality of student thinking and learning.


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ACKNOWLEDGEMENTS

This research was supported by a grant from the James S. McDonnell Foundation, Cognitive Studies in

Education Practice, grant no. JSMF 95-11. We wish to acknowledge the contributions of Hermi

Tabachneck who participated in early design and analysis and Ben MacLaren who did the solution

coding. Thanks also to Martha Alibali, Lisa Haverty, and Heather McQuaid for reading and

commenting on drafts and to Allan Collins, Andee Rubin, and an anonymous reviewer for their helpful

comments.


FIGURES

Figure 1. Quantitative problem solving involves two phases, comprehension and solution, both of

which are influenced by the external representation (e.g., story, word, equation) in which a problem is

presented. The influence on the comprehension phase results from the need for different kinds of

linguistic processing knowledge (e.g., situational, verbal, or symbolic) required by different external

representations. The impact on the solution phase results from the different computational

characteristics of the strategies (e.g., unwind, guess-and-test, equation solving) cued by different

external representations.


Situation &Problem Models

Problem Presentation Solution Notations

InternalProcessingExternal

Representations

ComprehensionPhase

KnowledgeSources

Symbolic

SolutionPhase

Unwind

EquationSolving

SolutionStrategies

8. After buyingdonuts at WholeyDonuts, Lauramutiplies the numberof donuts she boughtby their price of $0.37per donut. Then sheadds the $0.22charges for the boxthey came in and gets$2.81. How manydonuts did she buy?

5. Starting withsome number, if Imultiply it by .37and then add .22,I get 2.81. Whatnumber did I startwith?

2. Solve for x: x * .37 + .22 = 2.81

$.37

$.22

$2.81

+

*

X

Guess& Test

Situational

Verbal

$$


Whole Number Problems Decimal Number Problems

Figure 2. Proportion correct of high school algebra students in DFA1 (n = 76). The graphs show the

effects of the three difficulty factors: representation, unknown-position, and number-type. The error

bars display standard errors around the item means.

Result-Unknown Start-Unknown0

.1

.2

.3

.4

.5

.6

.7

.8

.91

EquationWord-EquationStory

Unknown PositionResult-Unknown Start-Unknown

0.1.2

.3

.4

.5

.6

.7

.8

.91


Unknown Position


Whole Number Problems Decimal Number Problems

Figure 3. Proportion correct of high school algebra students in DFA2 (n = 171). The graphs show the

effects of the three difficulty factors: representation, unknown-position, and number-type. The error

bars display standard errors around the item means.


.1

.2

.3

.4

.5

.6

.7

.8

.91


Unknown Position


.1

.2

.3

.4

.5

.6

.7

.8

.91


Unknown Position


a. The normative strategy: Translate to algebra and solve algebraically

b. The guess-and-test strategy.

c. The unwind strategy.


d. Translation to an algebra equation, which is then solved by the informal, unwind strategy.

e. A rare translation of a result-unknown story to an equation.

Figure 4. Examples of successful strategies used by students: (a) Guess-and-test, (b) Unwind, (c)

Translate to algebra and solve algebraically, (d) Translate to algebra and solve by unwind, (e) Translate

to algebra and solve by arithmetic


a. Order of operations error. Student inappropriately adds .37 and .22.

b. Algebra manipulation errors. Student subtracts from both sides of the plus sign rather than both sides

of the equal sign.

c. Argument order error . Student treats "subtract 66" (x – 66) as if it were "subtract from 66" (66 – x).


d. Inverse operator error. Student should have added .10 rather than subtracting .10.

e. Incomplete guess-and-test error. Student gives up before finding a guess that works.

f. Decimal alignment arithmetic error. Student adds 66 to .90 aligning flush right rather than aligning

place values or the decimal point.

Figure 5. Examples of errors made by students: (a) order of operations, (b) algebra manipulation, (c)

argument order, (d) inverse operator, (e) incomplete guess-and-test, (f) decimal alignment arithmetic

error


(a) Proportions of correct and incorrect responses for story, word equations, and equations. A verbal

facilitation is indicated particularly in the fewer no-response errors in the story and word equation

problems.


(b) Proportions of correct and incorrect responses for story vs. word equations crossed with whole vs.

decimal number problems. A situation facilitation effect is indicated in decimal problems with fewer

conceptual and arithmetic errors on story decimal problems than word-equation decimal problems

Figure 6. Frequency of broad error categories helps to explain the causes of the verbal and situation

facilitation effects observed.


TABLES

Table 1. Six Problem Categories Illustrating Two Difficulty Factors: Representation and Unknown-Position.

STORY PROBLEM WORD EQUATION SYMBOLIC EQUATION

RESULT-UNKNOWN

When Ted got home from his waiter job, he took the $81.90 he earned that day and subtracted the $66 he received in tips. Then he divided the remaining money by the 6 hours he worked and found his hourly wage. How much does Ted make per hour?

Starting with 81.9, if I subtract 66 and then divide by 6, I get a number. What is it?

Solve for x: (81.90 - 66)/6 = x

START-UNKNOWN

When Ted got home from his waiter job, he multiplied his hourly wage by the 6 hours he worked that day. Then he added the $66 he made in tips and found he had earned $81.90. How much does Ted make per hour?

Starting with some number, if I multiply it by 6 and then add 66, I get 81.9. What number did I start with?

Solve for x: x * 6 + 66 = 81.90


Table 2. Examples of the Four Cover Stories Used.

STORY PROBLEM COVER STORIES

DONUT LOTTERY WAITER BASKETBALL

After buying donuts at Wholey Donuts, Laura multiplies the 7 donuts she bought by their price of $0.37 per donut. Then she adds the $0.22 charge for the box they came in and gets the total amount she paid. How much did she pay?

After hearing that Mom won a lottery prize, Bill took the $143.50 she won and subtracted the $64 that Mom kept for herself. Then he divided the remaining money among her 3 sons giving each the same amount. How much did each son get?

When Ted got home from his waiter job, he multiplied his wage of $2.65 per hour by the 6 hours he worked that day. Then he added the $66 he made in tips and found how much he earned. How much did Ted earn that day?

After buying a basketball with his daughters, Mr. Jordan took the price of the ball, $68.36, and subtracted the $25 he contributed. Then he divided the rest by 4 to find out what each daughter paid. How much did each daughter pay?


Number type -> Integer Decimal

Cover story -> Donut Lottery Waiter Bball Donut Lottery Waiter BballBase equation -> 4*25+10

= 11020*3+40

= 1004*6 + 66

= 903*5 + 34

= 497 * .37

+ .22= 2.81

26.50* 3+ 64

= 143.50

2.65* 6+ 66

= 81.90

10.84*4+ 25

= 68.36Presenta

-tionUnk. posit.

Final arith

story result *, + 1.1story result -, / 1.3story start *, + 1.4story start -, / 1.2

word-eq result *, +word-eq result -, / 1.6word-eq start *, +word-eq start -, / 1.5equation result *, + 1.8equation result -, /equation start *, + 1.7equation start -, /

Table 3. Difficulty Factor Space and Distribution of Problems on Forms


Table 4. Solution strategies (%'s) employed by solvers as a function of problem representation for start-unknown problems.

Problem representation

Unwind Guess & Test

Symbol Manipulation

No Response

Answer Only

Unknown Total

Story 50 7 5 12 18 8 100

Word equation 22 23 11 13 19 12 100

Equation 13 14 22 32 11 8 100


Table 5. Likelihood that strategies used by students on start-unknown problems leads to a correct answer.

Strategies No. correct / no.

attempted

Likelihood strategy leads

to correct answer

Unwind 232 /335 69%

Guess & Test 89 / 125 71%

Symbol Manipulation 56 / 109 51%

Answer Only 90/ 161 56%

Unknown 38/ 89 43%

Total with Response 505 / 819 62%

No Response 0 / 169 0%

Total 505 / 988 51%

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